The onset of instability in a liquid annular jet enclosing another fluid, and surrounded by a gas in a pipe is analysed by use of a spectra-collocation method. The dynamic responses to the variation of different flow parameters are elucidated by use of numerical results. Two linearly independent convectively unstable interfacial modes of disturbances are found. In general, the para-sinuous mode has a larger amplification rate than the para-varicose mode. It is shown that to initiate encapsulation of core fluid with a uniform shell fluid, the growth of the para-sinuous mode must be promoted and the para-varicose mode must be suppressed. Suppression of the paravaricose mode in a finite range of wavenumbers is possible by varying the flow parameters. The effects of ten relevant parameters on instability are discussed. In certain parameter space, the annular jet becomes absolutely unstable with respect to the sinuous mode. The transition Weber number below which the flow is absolutely unstable and above which the flow is convectively unstable is found as a function of the Reynolds number when the rest of flow parameters are given. A successful encapsulation of core fluid with a uniform shell fluid is possible if the process is carried out outside of the parameter space of absolute instability, and if an external forcing is introduced at a frequency within a band in which the para-varicose mode is stable but the para-sinuous mode is convectively unstable.